Shimura curves contained in the Jacobian locus in small genus

Samuel Grushevsky, February 28th, 2014

Shimura subvarieties of the moduli space of polarized abelian varieties are defined
from some number theoretic data. The locus of Jacobians of curves is a geometrically
defined subvariety of the moduli space of principally polarized abelian varieties. Thus
the natural question of describing Shimura subvarieites of the Jacobian locus
intertwines the questions of number theory and algebraic geometry, and in fact it is
expected that in sufficiently high dimension/genus the Jacobian locus contains no
Shimura varieties. In contrast, in genus 4 we construct infinitely many Shimura curves
contained in the Jacobian locus, and in genus 3 we construct infinitely many Shimura
curves contained in the locus of hyperelliptic Jacobians. Based on joint work with
Martin Moeller.